Polynomial representations applications

Energy-Separable Polynomial Representation of the Time-Independent Full Green Operator with Application to Time-Independent Wavepacket Forms of Schrodinger and Lippmann-Schwinger Equations. 

Energy-Separable Faber Polynomial Representation of Operator Functions Theory and Application in Quantum Scattering. 

Huang Y., Kouri, D.J. and Hoffman, D.K. (1994) General, energy-separable Faber polynomial representation of operator-functions - theory and application in quantum scattering J. Chem. Phys. 101, 10493-10506. 

State calculations. With the extensions provided, the method can be applied to the full Watson Hamiltonian [51] for the vibrational problem. The efficiency of the method depends greatly on the nature of the anharmonic potential that represents couphng between different vibrational modes. In favorable cases, the latter can be represented as a low-order polynomial in the normal-mode displacements. When this is not the case, the computational effort increases rapidly. The Cl-VSCF is expected to scale as or worse with the number N of vibrational modes. The most favorable situation is obtained when only pairs of normal modes are coupled in the terms of the polynomial representation of the potential. The VSCF-Cl method was implemented in MULTIMODE [47,52], a code for anharmonic vibrational spectra that has been used extensively. MULTIMODE has been successfully applied to relatively large molecules such as benzene [53]. Applications to much larger systems could be difficult in view of the unfavorable scalability trend mentioned above. 

Y. Huang, D. J. Kouri, and D. K. Hoffman, A general, energy-separable polynomial representation of the time-independent full Green operator with application to time-independent wavepacket forms of Schrodinger and Lippmann-Schwinger equations, Chem. Phys. Lett. 225 31 (1994). 

This chapter is intended to present an integrated description of this general approach to quantum dynamics. Applications of the equations and strategies both to scattering and bound state problems will be discussed. In the next section, we begin with a detailed summary of the salient features of the DAFs as they are used to represent the Hamiltonian operator. Then in Sec. Ill, we discuss the TIWSE and some of the choices that can be made in solving for bound states and scattering information. Included in this is a discussion of the polynomial representations of various operators involved in the TIW form of quantum mechanics. Finally, in Sec. IV we briefly summarize some of the applications made to date of this overall approach. 

Modification of a mesh can be best automated by the application of the novel parametric mesh. The polynomial representation of a mesh can be modified by changed parameters. Typical or task related values of parameters can be stored in databases. 

As mentioned in Section IV. A, a straightforward way to deal with optimal control problems is to parameterize them as piecewise polynomial functions on a predefined set of time zones. This suboptimal representation has a number of advantages. First, the approaches developed in the previous subsection can be applied directly. Secondly, for many process control applications, control moves are actually implemented as piecewise constants on fixed time intervals, so the parameterization is adequate for this application. 

In brief, the basis of the finite element method is the representation of a body or a structure by an assemblage of subdivisions called finite element. Simple fimctions are chosen approximate the distribution or variation of the actual displacements over each finite element. The unknown magnitudes or amplitudes of the displacement fimctions are the displacements at the nodal points. Hence, the final solution will yield the approximate displacements at the discrete locations in the body, the nodal points. A displacement function can be expressed in various forms, such as polynomials and trigonometric fimctions. Since polynomials offer ease in mathematical manipulations, they have been employed in finite element applications. 

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